Problem 13 - Complex Analysis

Question

Sketch the following level lines for the map \(f: \mathbb{C} \rightarrow \mathbb{C}, z \mapsto z^{3}\) $$ \\{z \in \mathbb{C} ; \operatorname{Re} f(z)=c\\}, \quad\\{z \in \mathbb{C} ; \operatorname{Im} f(z)=c\\}, \quad\\{z \in \mathbb{C} ;|f(z)|=c\\} $$ for \(c \in \mathbb{Z}\) with \(|c| \leq 5\). Go on to find the images under \(f\) of these level lines and the images of the lines parallel to the real axis (resp. the imaginary axis).

Step-by-Step Solution

Answer

The level sets create complex curves and concentric circles; their images under the map are transformed in rotation, scaling, and positioning in the complex plane.

Step 1: Understanding the Function

The function given is \( f: \mathbb{C} \rightarrow \mathbb{C}, z \mapsto z^{3} \). This means that for any complex number \( z \), the function outputs \( z^{3} \).

Step 2: Sketch Level Lines for \( \operatorname{Re} f(z) = c \)

The condition \( \operatorname{Re}(f(z)) = c \) means we consider points such that the real part of \( z^3 = c \). Factorize \( z \) as \( z = re^{i\theta} \), then \( z^3 = r^3e^{3i\theta} \). The real part is \( r^3 \cos(3\theta) = c \). For each \( c \), solve \( \cos(3\theta) = \frac{c}{r^3} \) and plot the corresponding curves, known as folia or trisectrix type patterns.

Step 3: Sketch Level Lines for \( \operatorname{Im} f(z) = c \)

The condition \( \operatorname{Im}(f(z)) = c \) means we consider points such that the imaginary part of \( z^3 = c \). The imaginary part is \( r^3 \sin(3\theta) = c \). For each integer \( c \) such that \(|c| \leq 5\), solve \( \sin(3\theta) = \frac{c}{r^3} \) and plot the curves forming similar patterns to those found for the real part.

Step 4: Sketch Level Lines for \(|f(z)| = c\)

The condition \(|f(z)| = c\) takes the modulus of \(z^3\) equal to \(c\). This translates to \(|z^3| = |z|^3 = c\), which means \( |z| = c^{1/3} \). For each integer value of \( |c| \leq 5 \), draw circles of radius \( c^{1/3} \) around the origin.

Step 5: Find Image of Level Lines for \( \operatorname{Re} f(z) = c \)

The image of lines for \( \operatorname{Re} z = \text{constant} \) under \(f(z) = z^3\) rotates these lines due to the cubic transformation, producing curves along lines where the real part of \(z^3\) is \(3c\).

Step 6: Find Image of Level Lines for \( \operatorname{Im} f(z) = c \)

The image of lines for \( \operatorname{Im} z = \text{constant} \) similarly rotates through \(z^3\), producing a complex set of curves, with the imaginary part becoming \(3c\).

Step 7: Find Image of |f(z)| = c

Since the level set forms concentric circles, the image under \(z^3\) is also a set of concentric circles but expanded by a factor of 3 in absolute value, so \(|f(z)| = c^3\).

Step 8: Image of Lines Parallel to the Real and Imaginary Axes

Lines parallel to the real axis or the imaginary axis will transform to curves with rotated and stretched or compressed versions of these features, frequently forming spirals.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane used to visualize and operate with complex numbers. It comprises a horizontal real axis and a vertical imaginary axis. Each complex number is represented as a point in this plane, where the horizontal component is the real part, and the vertical component is the imaginary part. For example, the complex number \[ z = x + yi \] is represented as the point \( (x, y) \).

• The x-coordinate represents the real part of the complex number.
• The y-coordinate represents the imaginary part.

Visualizing the complex numbers on this plane helps us understand operations involving complex numbers, such as addition and multiplication. These operations involve both changing magnitude and rotating points on the plane.

Level Curves

In the context of complex functions, level curves are paths along which the function takes a constant value. There are different types of level curves in the complex plane:

• Re(f(z)) = c: These are curves where the real part of the function output is constant. For a function like \( f(z) = z^3 \), these level curves could create intricate patterns due to the nature of how complex numbers map onto themselves.
• Im(f(z)) = c: Similar to the real part, these curves represent locations where the imaginary part is constant.
• |f(z)| = c: The modulus condition results in circles whose radius is determined by the constant value of the modulus.

Level curves provide a beautiful insight into the behavior of complex functions and help in visualizing how different inputs relate to each other through the function.

Conformal Mapping

Conformal mapping is a crucial concept in complex analysis, where a function preserves angles but may alter distances between points. This property makes conformal mappings especially useful in applications such as fluid dynamics, where maintaining angles is critical.

• Conformal maps preserve the structure of small shapes while possibly transforming them at a larger scale.
• The function \( f(z) = z^3 \) is an example of a conformal map except at the origin, where it rotates and scales shapes in the complex plane.

In the context of the exercise, when we apply \( z^3 \) to curves defined with level lines for real and imaginary parts, we achieve distorted but angle-preserving images of the original shapes. Understanding how these transformations work helps in solving problems involving complex mappings.

Complex Analysis

Complex analysis is a fundamental area of mathematics focused on functions that operate on complex numbers. This field explores how complex functions behave, including properties like differentiability and integration. Some essential aspects of complex analysis include:

• Continuity and differentiability of complex functions: While similar to real analysis, complex differentiability involves holomorphic functions, which are more restrictive but rich in properties.
• The Cauchy-Riemann equations: These conditions ensure that a function is differentiable in a complex sense.
• Applications of complex analysis: From solving potential flow in physics to providing insights in electrical engineering, this field has vast applications across science and technology.

In our given exercise, complex analysis provides the tools to understand the transformations and behavior of the mappings from \( f(z) = z^3 \). By exploring these transformations, students gain deeper insights into the mathematical and visual nature of complex functions.